#4386278

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(7,10)(8,12)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30,34,35)(31,36,38,33)(32,37,40,42)(39,43,44,41);(5,6,9,11)(7,8,10,12)(15,21)(19,22)(23,27)(26,28)(30,35)(31,38)(32,40)(41,43);(1,2,3,4)(5,7,9,10)(6,8,11,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,31)(30,36)(32,39)(33,35)(34,38)(37,43)(40,44)(41,42);(5,8)(6,10)(7,11)(9,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32)(30,37)(31,39)(33,41)(34,40)(35,42)(36,43)(38,44);(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,33,34,36)(30,38,35,31)(32,41,40,43)(37,44,42,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4293474);